The Spectrum of a Glued Matrix

A glued matrix can be obtained from a direct sum of $p$ copies of an unreduced symmetric tridiagonal matrix $T$ by modifying the junctions by a glue $\gamma$, in one of two ways, so that the new tridiagonal matrix has no zero off-diagonal entries. Despite being unreduced, a glued matrix can have some eigenvalues agreeing to hundreds of decimal places. This makes glued matrices practically useful as test matrices for tridiagonal eigensolvers such as inverse iteration and the MRRR algorithm. However, the eigenvalue distribution of a glued matrix is a fascinating subject of theoretical interest in its own right. By means of secular equations, this paper studies how width and placement of the eigenvalue clusters of a glued matrix depend on $T$, on $p$, and on $\gamma$. Interlacing properties and the question of eigenvalue repetition between $T$ and a glued matrix are also investigated.